Solving the Problem of Fuzzy Partition-Distribution with Determination of the Location of Subset Centers

Abstract

A large number of real-world problems from various fields of human activity can be reduced to optimal partitioning-allocation problems with the purpose of minimizing the partitioning quality criterion. A typical representative of such problem is an infinite-dimensional transportation problem and more generalized problems—infinite-dimensional problems of production centers placement along with the partitioning of the area of continuously distributed consumers with the purpose of minimizing transportation and production costs. The relevant problems are characterized by some kind of uncertainty level of a not-probabilistic nature. A method is proposed to solve an optimal fuzzy partitioning-allocation problem with the subsets centers placement for sets of n-dimensional Euclidean space. The method is based on the synthesis of the methods of fuzzy theory and optimal partitioning-allocation theory, which is a new science field in infinite-dimensional mathematical programming with Boolean variables. A theorem was proved that determines the form of the optimal solution of the corresponding optimal fuzzy partitioning-allocation problem with the subsets centers placement for sets of n-dimensional Euclidean space. An algorithm for solving fuzzy partitioning-allocation problems is proposed, which is based on the proved theorem and on a variant of Shor’s r-algorithm—a non-differential optimization method. The application of the proposed method is demonstrated on model tasks, where the coefficient of mistrust is integrated to interpret the obtained result—the minimum value of the membership function, which allows each point of the set partition to be assigned to a specific fuzzy subset.